x^2+78x=729

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Solution for x^2+78x=729 equation:


Simplifying
x2 + 78x = 729

Reorder the terms:
78x + x2 = 729

Solving
78x + x2 = 729

Solving for variable 'x'.

Reorder the terms:
-729 + 78x + x2 = 729 + -729

Combine like terms: 729 + -729 = 0
-729 + 78x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '729' to each side of the equation.
-729 + 78x + 729 + x2 = 0 + 729

Reorder the terms:
-729 + 729 + 78x + x2 = 0 + 729

Combine like terms: -729 + 729 = 0
0 + 78x + x2 = 0 + 729
78x + x2 = 0 + 729

Combine like terms: 0 + 729 = 729
78x + x2 = 729

The x term is 78x.  Take half its coefficient (39).
Square it (1521) and add it to both sides.

Add '1521' to each side of the equation.
78x + 1521 + x2 = 729 + 1521

Reorder the terms:
1521 + 78x + x2 = 729 + 1521

Combine like terms: 729 + 1521 = 2250
1521 + 78x + x2 = 2250

Factor a perfect square on the left side:
(x + 39)(x + 39) = 2250

Calculate the square root of the right side: 47.434164903

Break this problem into two subproblems by setting 
(x + 39) equal to 47.434164903 and -47.434164903.

Subproblem 1

x + 39 = 47.434164903 Simplifying x + 39 = 47.434164903 Reorder the terms: 39 + x = 47.434164903 Solving 39 + x = 47.434164903 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-39' to each side of the equation. 39 + -39 + x = 47.434164903 + -39 Combine like terms: 39 + -39 = 0 0 + x = 47.434164903 + -39 x = 47.434164903 + -39 Combine like terms: 47.434164903 + -39 = 8.434164903 x = 8.434164903 Simplifying x = 8.434164903

Subproblem 2

x + 39 = -47.434164903 Simplifying x + 39 = -47.434164903 Reorder the terms: 39 + x = -47.434164903 Solving 39 + x = -47.434164903 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-39' to each side of the equation. 39 + -39 + x = -47.434164903 + -39 Combine like terms: 39 + -39 = 0 0 + x = -47.434164903 + -39 x = -47.434164903 + -39 Combine like terms: -47.434164903 + -39 = -86.434164903 x = -86.434164903 Simplifying x = -86.434164903

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.434164903, -86.434164903}

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